

The results follow from (REF ) and the facts that
\(f^\uparrow _C(z,v) \le f^{1\uparrow }_C(z,v)+f^{2\uparrow }_C(z,v)\)  [1]} and \(f^\downarrow _C(z,v) =-f^\uparrow _C(z,-v)\)  [1]}, \(\forall z \in \mathcal {Z}, \forall v \in \mathbb {R}^{n_z}\) .

